CS5050 Homework 5 Chapter 10 and test review

CS5050 Homework 5
Chapter 10 and test review
1. Consider a Monte Carlo algorithm A for a problem P whose expected running time is at most T (n) on
any instance of size n and that produces a correct solution with probability fl(n). Suppose further that
given a solution to P , we can verify its correctness in time t(n). Show how to obtain a Las Vegas
Algorithm that always gives a correct answer to P and runs in expected time at most (T(n) + t(n))/fl(n).
2. Liz conjectures that for any n, n-1 is its own inverse mod n. Prove or disprove this conjecture.
3. Discuss the relationship between the FastExponentiation (page 463) for n k and the binary representation
for k.
4. Show the execution of ExtendedEuclidGCD(412,113) by constructing a table similar to Table 10.10
(page 465).
5. Prove Corollary 10.7: if gcd(x,n) = 1 then Zn={ix mod n:i=0,1,...,n-1}
6. (C-10.4) Show how to modify algorithm ExtendedEuclidGCD to compute the multiplicative inverse of
an element in Zn using arithmetic operations on operands with at most 2log n bits.
7. Compute successive powers for the elements of Z17 in a table similar to that of Table 10.5 (page 460).
8. Compute the multiplicative inverses of 113, 114, and 127 in Z299.
9. Construct the modulo multiplication table for Z11
* 0 1 2 3 4 5 6 7 8 9 10
0
1
2
3 7
4
5 9
6
7
8
9
10
10. Consider the following algorithm for GCD:
int binGCD(int a,int b)
{ if (a == 0) return b;
if (b == 0) return a;
if (a%2==0 && b % 2 = 0) return 2*binGCD(a/2,b/2);
if (a%2==1 && b % 2 = 0) return binGCD(a,b/2);
if (a%2==0 && b % 2 = 1) return binGCD(a/2,b);
return binGCD(Math.abs(a-b)/2, b);
}
A friend says, ``I don't like it. When I call binGCD(3,15), it creates a call
to binGCD(6,15). This makes me nervous because we have turned a smaller
problem into a larger problem. I was taught recursion shouldn't do that?''
How do you answer your friend?

11. For any integers a and b, if d|a and d|b then d|gcd(a,b) . Prove this.
12. If p if a prime, the inverse of x mod p is given by:
(a)
(b)
(c)
(d) there may not be an inverse for x
13. For n = 12, what is Φ(n)?
(a) 2 (b) 3 (c) 4 (d) 6 (e) 12
14. For n = 17*23, what is Φ(n) ?
(a) 17 (b) 45 (c) 40 (d) 352 (e) 390
15. Which is true of the Boyer-Moore Algorithm for matching pattern P in text T?
a. the algorithm begins searching at the end of the pattern string
b. if the mismatch occurs at T[i] = c and c occurs nowhere in P, the entire pattern is shifted
past T[i].
c. if the mismatch occurs and T[i] = c and c occurs at location L[c], you only move one
location, if moving to L[c] would shift the pattern backwards.
d. all of the above
e. none of the above
16. In constructing the last function for the Boyer-Moore Algorithm in which the pattern length is
m and the total alphabet is A, the complexity would be
a. O(A)
b. O(m+A)
c. O(mA)
d. O(1)
17. In a standard trie containing n strings, the height of the trie is
a. log n
b. the length of the longest string
c. slightly more than the height of the average string
d. no formula for height is known, as it depends on the specific data
e. none of the above
18. The binary failure table (assuming a binary input string) for the string ddedddeeded as
follows:

a.
b.
c.
d.
e. None of the above
19. We want to find two numbers - one which is smaller than the median and one which is larger
than the median. We pick three numbers. We claim the biggest is above the median and the
smallest is below the median. What is the probability we have made an error?
(a) .05 (b) .125 (c) .25 (d) .50 (e) 1
20. Given any number x, the easiest way to find its inverse mod n is
a. build a table of powers of x and stop when you get to a 1.
b. build a multiplication table for Zn and look for a 1 in the x column or row.
c. use extendedGCD(x,n) and use xmult as the possible inverse.
d. if gcd(x,n) = 1, the inverse is 1/x.
e. none of the above